Nonlinear Controllability and the Propagation of Local Information: From the Kalman Family to Lie Brackets, Rotation Groups, and Reachable Subgroups
arXiv:2608.20094
2026
Architecture
2 ideas extracted · analyzed Sep 1, 2026
What the math gives to ML
The paper provides a constructive finite-dimensional mechanism for propagating a small set of local control directions through Lie brackets. In matrix Lie groups, repeated applications of the adjoint map \(\operatorname{ad}_A(C)=[A,C]\) generate new directions analogously to the linear controllability family \(B,AB,A^2B,\ldots\), while products of matrix exponentials implement these directions as exactly orthogonal transformations. This can be transferred into neural networks as a parameter-efficient orthogonal mixing layer: learn a few skew-symmetric generators and synthesize additional channel rotations through commutators or short control sequences. A second transferable use is a controllability-style regularizer that prevents the generated Lie directions from becoming linearly redundant.
Ideas from this paper
Unverified
2026
Replace a dense unconstrained channel-mixing matrix with a differentiable product of exponentials of a few skew-symmetric generators and their iterated commutators. The resulting layer is exactly orthogonal, preserves feature norms, and can express rotations in directions not explicitly stored as independent parameters. This is especially suitable for residual MLP blocks, recurrent state transitions, and networks processing rotation- or pose-valued features.
Useful6/10
Difficulty5/10
Novelty6/10
Unverified
2026
Regularize learned skew generators so that their iterated Lie brackets span many independent feature-mixing directions rather than collapsing to commuting or redundant matrices. This turns the paper's controllability family into a differentiable diversity objective for structured neural layers.
Useful5/10
Difficulty4/10
Novelty7/10